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# Permutation

In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting. [1]

114 relations: Algebra, Alternating permutation, Anagram, Ancient Society of College Youths, Array data structure, Associative property, Augustin-Louis Cauchy, Évariste Galois, Bhāskara II, Big O notation, Bijection, Binomial coefficient, Bubble sort, Cardinality, Cayley table, Cayley's theorem, Change of variables, Change ringing, Combination, Combinatorics, Commutative property, Computer science, Conjugacy class, Convolution, Cycle index, Cyclic order, Cyclic permutation, Derangement, Disjoint sets, Dominique Foata, Donald Knuth, Ernő Rubik, Error detection and correction, Eulerian number, Fabian Stedman, Factorial, Factorial number system, Finite set, Fixed point (mathematics), Forward error correction, Frank Yates, Function (mathematics), Function composition, Galois theory, Group (mathematics), Group action, Group representation, Group theory, Heap's algorithm, Heinrich August Rothe, ... Expand index (64 more) »

## Algebra

Algebra (from Arabic and Farsi "al-jabr" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.

## Alternating permutation

In combinatorial mathematics, an alternating permutation of the set is an arrangement of those numbers into an order c1,..., cn such that no element ci is between ci &minus; 1 and ci + 1 for any value of i and c1&lt; c2.

## Anagram

An anagram is a type of word play, the result of rearranging the letters of a word or phrase to produce a new word or phrase, using all the original letters exactly once; for example, the word anagram can be rearranged into nag-a-ram.

## Ancient Society of College Youths

The Ancient Society of College Youths (ASCY) is the world's premier society of church bellringers, founded in 1637 and based in the City of London.

## Array data structure

In computer science, an array data structure or simply an array is a data structure consisting of a collection of elements (values or variables), each identified by at least one array index or key.

## Associative property

In mathematics, the associative property is a property of some binary operations.

## Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy FRS FRSE (21 August 1789 – 23 May 1857) was a French mathematician reputed as a pioneer of analysis.

## Évariste Galois

Évariste Galois (25 October 1811 – 31 May 1832) was a French mathematician born in Bourg-la-Reine.

## Bhāskara II

Bhāskara (also known as Bhāskarācārya ("Bhāskara the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I) (1114–1185), was an Indian mathematician and astronomer.

## Big O notation

In mathematics, big O notation describes the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions.

## Bijection

In mathematics, a bijection, bijective function or one-to-one correspondence is a function between the elements of two sets, where every element of one set is paired with exactly one element of the other set, and every element of the other set is paired with exactly one element of the first set.

## Binomial coefficient

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem.

## Bubble sort

Bubble sort, sometimes referred to as sinking sort, is a simple sorting algorithm that repeatedly steps through the list to be sorted, compares each pair of adjacent items and swaps them if they are in the wrong order.

## Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

## Cayley table

A Cayley table, after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table.

## Cayley's theorem

In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G. This can be understood as an example of the group action of G on the elements of G. A permutation of a set G is any bijective function taking G onto G; and the set of all such functions forms a group under function composition, called the symmetric group on G, and written as Sym(G).

## Change of variables

In mathematics, the operation of substitution consists in replacing all the occurrences of a free variable appearing in an expression or a formula by a number or another expression.

## Change ringing

Change ringing is the art of ringing a set of tuned bells in a series of mathematical patterns called "changes".

## Combination

In mathematics, a combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter.

## Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.

## Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

## Computer science

Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations Computer science is the scientific and practical approach to computation and its applications.

## Conjugacy class

In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure.

## Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated.

## Cycle index

In combinatorial mathematics a cycle index is a polynomial in several variables which is structured in such a way that information about how a group of permutations acts on a set can be simply read off from the coefficients and exponents.

## Cyclic order

In mathematics, a cyclic order is a way to arrange a set of objects in a circle.

## Cyclic permutation

In mathematics, and in particular in group theory, a cyclic permutation is a permutation of the elements of some set X which maps the elements of some subset S of X to each other in a cyclic fashion, while fixing (i.e., mapping to themselves) all other elements of X. For example, the permutation of that sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1 is a cycle, while the permutation that sends 1 to 3, 3 to 1, 2 to 4 and 4 to 2 is not (it separately permutes the pairs and). A cycle in a permutation is a subset of the elements that are permuted in this way.

## Derangement

In combinatorial mathematics, a derangement is a permutation of the elements of a set, such that no element appears in its original position.

## Disjoint sets

In mathematics, two sets are said to be disjoint if they have no element in common.

## Dominique Foata

Dominique Foata (born October 12, 1934) is a mathematician who works in enumerative combinatorics.

## Donald Knuth

Donald Ervin Knuth (born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University.

## Ernő Rubik

Ernő Rubik (born 13 July 1944) is a Hungarian inventor, architect and professor of architecture.

## Error detection and correction

In information theory and coding theory with applications in computer science and telecommunication, error detection and correction or error control are techniques that enable reliable delivery of digital data over unreliable communication channels.

## Eulerian number

In combinatorics, the Eulerian number A(n, m), is the number of permutations of the numbers 1 to n in which exactly m elements are greater than the previous element (permutations with m "ascents").

## Fabian Stedman

Fabian Stedman (b. Yarkhill, Herefordshire 1640, d. 1713) was a leading figure in campanology and bell-ringing.

## Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The value of 0! is 1, according to the convention for an empty product.

## Factorial number system

In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations.

## Finite set

In mathematics, a finite set is a set that has a finite number of elements.

## Fixed point (mathematics)

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function.

## Forward error correction

In telecommunication, information theory, and coding theory, forward error correction (FEC) or channel coding is a technique used for controlling errors in data transmission over unreliable or noisy communication channels.

## Frank Yates

Frank Yates FRS (May 12, 1902 &ndash; June 17, 1994) was one of the pioneers of 20th century statistics.

## Function (mathematics)

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

## Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

## Galois theory

In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements together with an operation that combines any two elements to form a third element.

## Group action

In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object.

## Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.

## Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

## Heap's algorithm

Heap's algorithm generates all possible permutations of objects.

## Heinrich August Rothe

Heinrich August Rothe (1773–1842) was a German mathematician, a professor of mathematics at Erlangen.

## Identity function

Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument.

## Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain.

## Insertion sort

Insertion sort is a simple sorting algorithm that builds the final sorted array (or list) one item at a time.

## Inverse function

In mathematics, an inverse function is a function that "reverses" another function.

## Inversion (discrete mathematics)

In computer science and discrete mathematics, an inversion is a pair of places of a sequence where the elements on these places are out of their natural order.

## Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism (or more generally a morphism) that admits an inverse.

## Joseph-Louis Lagrange

Joseph-Louis Lagrange, born Giuseppe Lodovico Lagrangia or Giuseppe Ludovico De la Grange Tournier (also reported as Giuseppe Luigi Lagrange or Lagrangia) (25 January 1736 &ndash; 10 April 1813) was an Italian Enlightenment Era mathematician and astronomer.

## Josephus problem

In computer science and mathematics, the Josephus Problem (or Josephus permutation) is a theoretical problem related to a certain counting-out game.

## Līlāvatī

The Līlāvatī is Indian mathematician Bhāskara II's treatise on mathematics, written in 1150.

## Lehmer code

In mathematics and in particular in combinatorics, the Lehmer code is a particular way to encode each possible permutation of a sequence of n numbers.

## Levi-Civita symbol

In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol represents a collection of numbers; defined from the sign of a permutation of the natural numbers 1, 2, …, n, for some positive integer n. It is named after the Italian mathematician and physicist Tullio Levi-Civita.

## Lexicographical order

In mathematics, the lexicographic or lexicographical order (also known as lexical order, dictionary order, alphabetical order or lexicographic(al) product) is a generalization of the way the alphabetical order of words is based on the alphabetical order of their component letters.

In computer science, a linked list is a data structure consisting of a group of nodes which together represent a sequence.

## List of order structures in mathematics

In mathematics, and more particularly in order theory, several different types of ordered set have been studied.

## List of permutation topics

This is a list of topics on mathematical permutations.

## LTE (telecommunication)

LTE, an abbreviation for Long-Term Evolution, commonly marketed as 4G LTE, is a standard for wireless communication of high-speed data for mobile phones and data terminals.

## Major index

In mathematics (and particularly in combinatorics), the major index of a permutation is the sum of the positions of the descents of the permutation.

## Mathematical induction

Mathematical induction is a method of mathematical proof typically used to establish a given statement for all natural numbers.

## Mathematics

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change.

## Matrix (mathematics)

In mathematics, a matrix (plural matrices) is a rectangular array—of numbers, symbols, or expressions, arranged in rows and columns—that is interpreted and manipulated in certain prescribed ways.

## Meander (mathematics)

In mathematics, a meander or closed meander is a self-avoiding closed curve which intersects a line a number of times.

## Miklós Bóna

Miklós Bóna (born on October 6, 1967, in Székesfehérvár) is an American mathematician of Hungarian origin.

Mixed radix numeral systems are non-standard positional numeral systems in which the numerical base varies from position to position.

## Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves the given order.

## Multiset

In mathematics, a multiset (or bag) is a generalization of the concept of a set that, unlike a set, allows multiple instances of the multiset's elements.

## Narayana Pandit

Narayana Pandita (নারায়ণ পণ্ডিত; नारायण पण्डित) (1340–1400) was a major mathematician of India.

## Necklace (combinatorics)

In combinatorics, a k-ary necklace of length n is an equivalence class of n-character strings over an alphabet of size k, taking all rotations as equivalent.

## Parity of a permutation

In mathematics, when X is a finite set of at least two elements, the permutations of X (i.e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations.

## Partial permutation

In combinatorial mathematics, a partial permutation, or sequence without repetition, on a finite set S is a bijection between two specified subsets of S. That is, it is defined by two subsets U and V of equal size, and a one-to-one mapping from U to V. Equivalently, it is a partial function on S that can be extended to a permutation.

## Partition (number theory)

In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers.

## Permutation

In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting.

## Permutation group

In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself).

## Permutation matrix

In mathematics, in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere.

## Permutation pattern

In combinatorial mathematics and theoretical computer science, a permutation pattern is a sub-permutation of a longer permutation.

## Permutation polynomial

In mathematics, a permutation polynomial (for a given ring) is a polynomial that acts as a permutation of the elements of the ring, i.e. the map x \mapsto g(x) is a bijection.

## Pochhammer symbol

In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation, where is a non-negative integer.

## Probability

Probability is the measure of the likeliness that an event will occur.

## Pseudocode

Pseudocode is an informal high-level description of the operating principle of a computer program or other algorithm.

## Q-Pochhammer symbol

In mathematics, in the area of combinatorics, a q-Pochhammer symbol, also called a q-shifted factorial, is a ''q''-analog of the common Pochhammer symbol.

## Random permutation

A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable.

## Rencontres numbers

In combinatorial mathematics, the rencontres numbers are a triangular array of integers that enumerate permutations of the set with specified numbers of fixed points: in other words, partial derangements.

## Representation theory of the symmetric group

In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained.

## Richard P. Stanley

Richard Peter Stanley (born June 23, 1944 in New York City, New York) is the Norman Levinson Professor of Applied Mathematics at the Massachusetts Institute of Technology, in Cambridge, Massachusetts.

## Robert Sedgewick (computer scientist)

Robert Sedgewick (born December 20, 1946) is a computer science professor at Princeton University and a member of the board of directors of Adobe Systems.

## Ronald Fisher

Sir Ronald Aylmer Fisher FRS (17 February 1890 – 29 July 1962), known as R.A. Fisher, was an English statistician, evolutionary biologist, mathematician, geneticist, and eugenicist.

## Scientific calculator

A scientific calculator is a type of electronic calculator, usually but not always handheld, designed to calculate problems in science, engineering, and mathematics.

## Sequence

In mathematics, a sequence is an ordered collection of objects in which repetitions are allowed.

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

## Sorting algorithm

A sorting algorithm is an algorithm that puts elements of a list in a certain order.

## Sorting network

In computer science, comparator networks are abstract devices built up of a fixed number of "wires", carrying values, and comparator modules that connect pairs of wires, swapping the values on the wires if they are not in a desired order.

A spreadsheet is an interactive computer application program for organization, analysis and storage of data in tabular form.

## Steinhaus–Johnson–Trotter algorithm

The Steinhaus–Johnson–Trotter algorithm or Johnson&ndash;Trotter algorithm, also called plain changes, is an algorithm named after Hugo Steinhaus, Selmer M. Johnson and Hale F. Trotter that generates all of the permutations of n elements.

## Stirling numbers of the first kind

In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations.

## String (computer science)

In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable.

## Substitution cipher

In cryptography, a substitution cipher is a method of encoding by which units of plaintext are replaced with ciphertext, according to a fixed system; the "units" may be single letters (the most common), pairs of letters, triplets of letters, mixtures of the above, and so forth.

## Superpattern

In the mathematical study of permutations and permutation patterns, a superpattern is a permutation that contains all of the patterns of a given length.

## Swap (computer science)

In computer programming, the act of swapping two variables refers to mutually exchanging the values of the variables.

## Symmetric group

In abstract algebra, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutation operations that can be performed on n distinct symbols, and whose group operation is the composition of such permutation operations, which are defined as bijective functions from the set of symbols to itself.

## The Art of Computer Programming

The Art of Computer Programming (sometimes known by its initials TAOCP) is a comprehensive monograph written by Donald Knuth that covers many kinds of programming algorithms and their analysis.

## Total order

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X which is transitive, antisymmetric, and total.

## Tuple

A tuple is a finite ordered list of elements.

## Turbo code

In information theory, turbo codes (originally in French Turbocodes) are a class of high-performance forward error correction (FEC) codes developed around 1990-91 (but first published in 1993), which were the first practical codes to closely approach the channel capacity, a theoretical maximum for the code rate at which reliable communication is still possible given a specific noise level.

## Twelvefold way

In combinatorics, the twelvefold way is a name given to a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number.

## Zero-based numbering

Zero-based numbering or index origin.

## References

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